Rational Ergodicity for Skew Products Cylinder Maps
Patricia Romano Cirilo
infinite ergodic theory | cylinder skew product | irrational rotation | ergodicity | rationally ergodic | weakly homogeneous.
In this thesis we study the asymptotic behavior of the ergodic Birkhoff Sums for cylinder skew products over irrational rotation preserving a σ-finite measure. We prove that such maps are ergodic, rationally ergodic and weakly homogeneous, calculating explicitly the Ergodic Sums for an increasing sequence of time and identifying the return sequence. From that, it is possible to obtain a second order ergodic theorem, which asserts that the double average renormalized by the return sequence converges to the integral of the observable function almost everywhere. We recall that the classical Birkhoff Theorem does not hold when the invariant measure is infinite.